Film only registers "Night" and "Day", which makes sense when you think about it. On camera, Sunset can't be distinguished from Sunrise. Only the sound effect of a rooster would give any inclination.

But I'm finding it an issue when parts of story unfold between the wee hours of "day light" and "darkness". For example, I have scene that takes place in the early morning (before actual sunrise). There's some light, but not enough to qualify the scene as "Day". So what should I do? Opinions on how that slugline should look?

You CAN use MORNING, DAWN, DUSK, LATE AFTERNOON, etc., though is generally preferred if you use Day or Night and then clarify/specify the particulars in the scene description like you did in your example.

The point of making the day/night designation is for production purposes. In your example, night is appropriate because the cast and crew would have to be setting up and shooting for hours before the sun rises if they are on location, or in the dark on a sound stage.